Is $x^5+x^4-2x+2$ irreducible in $\mathbb{Q}[x]$

Question

Is $x^5+x^4-2x+2$ irreducible in $\mathbb{Q}[x]$?

I think it is but I am only capable of showing that it has no roots in $\mathbb{Q}$.

Answer 1

This polynomial is irreducible over $\mathbf Z$, hence over $\mathbf Q$ because it is irreducible over $\mathbf Z/3\mathbf Z$, for the following reasons:

• It has no root in $\mathbf Z/3\mathbf Z=\{0, 1,-1\}$, so it has no linear factor.
• So if it has an irreducible (non trivial) factor, it has a quadratic irreducible factor.
• Now the irreducible quadratic polynomials in $\mathbf Z/3\mathbf Z[x]$ are $$\{x^2+1, x^2+x-1, x^2-x-1\}$$ (These are the only quadratic polynomials which have no root.)
• Performing the division of the given polynomial by each of these quadratic polynomials yields as a remainder $\; -x$, $1$, $1$ respectively.

Thus the polynomial has no linear nor quadratic irreducible factor, hence is irreducible.